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Connecting UV Luminosity to Stellar Mass

UV Luminosity Function

If we have a UV luminosity function, a connection
between UV luminosity and star formation rate,
and a relation between star formation rate and
stellar mass, we can estimate a stellar mass function.

Consider a UV luminosity function
with units Mpc mag,
such that the galaxy number density is

Star formation rate function

To convert the UV luminosity function into a star formation rate function, we need to have some connection between $M_{UV}$ and SFR.
Here, we will take to be the monochromatic UV luminosity
with units ergs s Hz. The connection between
star formation rate and is often written in the form

assuming that corresponds to the monochromatic frequency
that defines what we mean by “UV”. and
are constants. Throughout, I’ll implicitly divide by or such that the
arguments of logarithms are unit free.

We can work in magnitudes rather than luminosity by noting
that

where is some constant. If as above,
then we just have

or

or

We can then write the SFR function as

Stellar mass function

If we assume a one-to-one correlation between
star formation rate and stellar mass, then it’s straightforward
to convert between the SFR rate to the stellar mass function.

Let’s assume a connection between star formation rate and
stellar mass of

or

The units are , or no units!

We then have the stellar mass function as

This function has units of Mpc dex.

Scatter in stellar mass vs. SFR

Imagine we have a population of galaxies that have a distribution
of star formation rate and stellar mass. We could then write
that distribution as
such that

But we can relate the joint distribution of SFR and
stellar mass
to the conditional distribution of stellar mass given
star formation rate
as